In a light-colored saucepan or skillet over medium heat, melt the butter. Continue cooking, swirling the pan occasionally, for 5-6 minutes. It will foam and crackle before tiny brown specks (the "brown butter") form at the bottom, and it gives off a nutty aroma. Immediately pour into a heatproof bowl to stop the cooking. Let it cool for about 15-20 minutes, until it's no longer hot to the touch but still liquid.
In a large mixing bowl, whisk together the cooled brown butter, brown sugar, and white sugar until well combined. Add the egg, vanilla bean paste, and milk, then whisk until the mixture is smooth.
Add the flour, salt, and baking soda to the wet ingredients. Using a rubber spatula, fold the mixture until just a few streaks of flour remain. Add the chocolates, then continue folding until everything is well incorporated and no dry flour is visible (the dough should be slightly sticky).
Cover the bowl and refrigerate the dough for at least 20 minutes. This step is crucial for firming up the dough for shaping.
Using a cookie scoop (about 1.5-2 tablespoons) to portion the cookie and refrigerate the dough balls for another 30 minutes.
While the dough balls are chilling, preheat your oven to 180°C (356℉).
Arrange the chilled dough balls on the prepared baking sheet with parchment paper, spacing them at least 2-3 inches apart (6 per sheet is ideal). Bake for 13 minutes, or until the edges are set and golden brown, while the centers still look soft and slightly underdone.
Remove the pan from the oven. Let the cookies rest on the hot baking sheet for 10 minutes to finish setting and firm up. Then, carefully transfer them to a wire rack to cool completely.
